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Analysis of the “Toolkit” Method for the Time-Dependent Schrödinger Equation

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Abstract

The goal of this paper is to provide an analysis of the “toolkit” method used in the numerical approximation of the time-dependent Schrödinger equation. The “toolkit” method is based on precomputation of elementary propagators and was seen to be very efficient in the optimal control framework. Our analysis shows that this method provides better results than the second order Strang operator splitting. In addition, we present two improvements of the method in the limit of low and large intensity control fields.

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References

  1. Judson, R.S., Rabitz, H.: Teaching lasers to control molecules. Phys. Rev. Lett. 68, 1500 (1992)

    Article  Google Scholar 

  2. Assion, A., Baumert, T., Bergt, M., Brixner, T., Kiefer, B., Seyfried, V., Strehle, M., Gerber, G.: Control of chemical reactions by feedback-optimized phase-shaped femtosecond laser pulses. Science 282, 919–922 (1998)

    Article  Google Scholar 

  3. Levis, R.J., Menkir, G.M., Rabitz, H.: Selective bond dissociation and rearrangement with optimally tailored, strong-field laser pulses. Science 292, 709–713 (2001)

    Article  Google Scholar 

  4. Rabitz, H., de Vivie-Riedle, R., Motzkus, M., Kompa, K.: Wither the future of controlling quantum phenomena? Science 288, 824–828 (2000)

    Article  Google Scholar 

  5. Warren, W.S., Rabitz, H., Dahleh, M.: Coherent control of quantum dynamics: The dream is alive. Science 259, 1581–1589 (1993)

    Article  MathSciNet  Google Scholar 

  6. Weinacht, T.C., Ahn, J., Bucksbaum, P.H.: Controlling the shape of a quantum wavefunction. Nature 397, 233–235 (1999)

    Article  Google Scholar 

  7. Beauchard, K., Laurent, C.: Local controllability of 1d linear and nonlinear Schrödinger equations with bilinear control. J. Math. Pures Appl. 94(5), 520–554 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  8. Chambrion, T., Mason, P., Sigalotti, M., Boscain, U.: Controllability of the discrete-spectrum Schrödinger equation driven by an external field. Ann. Inst. Henri Poincare (C) Non Linear Anal. 26(1), 329–349 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  9. Nersesyan, V.: Global approximate controllability for Schrödinger equation in higher Sobolev norms and applications. Ann. Inst. Henri Poincare (C) Non Linear Anal. 27(3), 901–915 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  10. Chuang, I.L., Laflamme, R., Shor, P.W., Zurek, W.H.: Quantum computers, factoring, and decoherence. Science 270, 1633–1635 (1995)

    Article  MathSciNet  Google Scholar 

  11. Besse, C., Bidégaray, B., Descombes, S.: Order estimates in time of splitting methods for the nonlinear Schrödinger equation. SIAM J. Numer. Anal. 40(1), 26–40 (2002) (electronic)

    Article  MathSciNet  MATH  Google Scholar 

  12. Ciarlet, P.G., Lions, J.L. (eds.): Handbook of Numerical Analysis, vol. IX. North-Holland, Amsterdam (2003). Numerical methods for fluids. Part 3

    MATH  Google Scholar 

  13. Strang, G.: On the construction and comparison of difference schemes. SIAM J. Numer. Anal. 5, 506–517 (1968)

    Article  MathSciNet  MATH  Google Scholar 

  14. Yip, F.L., Mazziotti, D.A., Rabitz, H.: A local-time algorithm for achieving quantum control. J. Phys. Chem. A 107, 7264–7269 (2003)

    Article  Google Scholar 

  15. Yip, F.L., Mazziotti, D.A., Rabitz, H.: A propagation toolkit to design quantum control. J. Chem. Phys. 118(18), 8168–8172 (2003)

    Article  Google Scholar 

  16. Belhadj, M., Salomon, J., Turinici, G.: A stable toolkit method in quantum control. J. Phys. A: Math. Theor. 41, 362001 (2008)

    Article  MathSciNet  Google Scholar 

  17. Jahnke, T., Lubich, C.: Error bounds for exponential operator splittings. BIT Numer. Math. 40(4), 735–744 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  18. Sanz-Serna, J.M., Verwer, J.G.: Conservative and nonconservative schemes for the solution of the nonlinear Schrödinger equation. IMA J. Numer. Anal. 6(1), 25–42 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  19. Rabitz, H., Turinici, G., Brown, E.: Control of quantum dynamics: Concepts, procedures and future prospects. In: Ciarlet, Ph.G. (ed.) Computational Chemistry, Special Volume (C. Le Bris Editor) of Handbook of Numerical Analysis, vol. X, pp. 833–887. Elsevier, Amsterdam (2003)

    Google Scholar 

  20. Ben Haj-Yedder, A., Auger, A., Dion, C.M., Keller, A., Le Bris, C., Atabek, O.: Numerical optimization of laser fields to control molecular orientation. Phys. Rev. A 66, 063401 (2002)

    Article  Google Scholar 

  21. Turinici, G., Ramakhrishna, V., Li, B., Rabitz, H.: Optimal discrimination of multiple quantum systems: Controllability analysis. J. Phys. A, Math. Gen. 37(1), 273–282 (2003)

    Article  MathSciNet  Google Scholar 

  22. Turinici, G., Rabitz, H.: Optimally controlling the internal dynamics of a randomly oriented ensemble of molecules. Phys. Rev. A 70(6), 063412 (2004)

    Article  Google Scholar 

  23. Beauchard, K., Coron, J.-M., Rouchon, P.: Controllability issues for continuous-spectrum systems and ensemble controllability of Bloch equations. Commun. Math. Phys. 296, 525–557 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  24. Brezis, H.: Analyse fonctionnelle: Théorie et applications. Dunod, Paris (1994)

    MATH  Google Scholar 

  25. Cazenave, T., Haraux, A.: An Introduction to Semilinear Evolution Equations. Oxford University, London (1998)

    MATH  Google Scholar 

  26. Reed, M., Simon, B.: Methods of Modern Mathematical Physics, II, Fourier Analysis, Self-adjointness. Academic Press, San Diego (1975)

    MATH  Google Scholar 

  27. Sayood, K.: Introduction to Data Compression, 3rd edn. Morgan Kaufmann, San Mateo (2006)

    Google Scholar 

  28. Degani, I., Zanna, A., Saelen, L., Nepstad, R.: Quantum control with piecewise constant control functions. SIAM J. Sci. Comput. 31(5), 3566–3594 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  29. Balint-Kurti, G.G., Manby, F.R., Ren, Q., Artamonov, M., Ho, T., Rabitz, H.: Quantum control of molecular motion including electronic polarization effects with a two-stage toolkit. J. Chem. Phys. 122(8), 084110 (2005)

    Article  Google Scholar 

  30. Moler, C., Van Loan, C.: Nineteen dubious ways to compute the exponential of a matrix, twenty-five years later. SIAM Rev. 45(1), 3–49 (2003) (electronic)

    Article  MathSciNet  MATH  Google Scholar 

  31. Maday, Y., Rønquist, E.M.: A reduced-basis element method. In: Proceedings of the Fifth International Conference on Spectral and High Order Methods (ICOSAHOM-01), Uppsala, vol. 17, pp. 447–459 (2002)

    Google Scholar 

  32. Maday, Y., Rønquist, E.M.: The reduced basis element method: application to a thermal fin problem. SIAM J. Sci. Comput. 26(1), 240–258 (2004) (electronic)

    Article  MathSciNet  MATH  Google Scholar 

  33. Cancès, E., Le Bris, C., Maday, Y., Nguyen, N.C., Patera, A.T., Pau, G.S.H.: Feasibility and competitiveness of a reduced basis approach for rapid electronic structure calculations in quantum chemistry. In: High-Dimensional Partial Differential Equations in Science and Engineering. CRM Proc. Lecture Notes, vol. 41, pp. 15–47. Am. Math. Soc., Providence (2007)

    Google Scholar 

  34. Saad, Y.: Numerical Methods for Large Eigenvalue Problems. Manchester University Press, Manchester (1992)

    MATH  Google Scholar 

  35. Salomon, J., Dion, C.M., Turinici, G.: Optimal molecular alignment and orientation through rotational ladder climbing. J. Chem. Phys. 123(14), 144310 (2005)

    Article  Google Scholar 

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Authors and Affiliations

  1. CNRS, LAAS, 7 avenue du colonel Roche, 31077, Toulouse Cedex 4, France

    Lucie Baudouin

  2. Université de Toulouse, UPS, INSA, INP, ISAE, UT1, UTM, LAAS, 31077, Toulouse, France

    Lucie Baudouin

  3. Université Paris-Dauphine/CEREMADE, Pl. Mal. De Lattre de Tassigny, 75016, Paris, France

    Julien Salomon & Gabriel Turinici

Authors
  1. Lucie Baudouin
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  2. Julien Salomon
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  3. Gabriel Turinici
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Correspondence to Julien Salomon.

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Baudouin, L., Salomon, J. & Turinici, G. Analysis of the “Toolkit” Method for the Time-Dependent Schrödinger Equation. J Sci Comput 49, 111–136 (2011). https://doi.org/10.1007/s10915-010-9450-6

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  • DOI: https://doi.org/10.1007/s10915-010-9450-6

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